A NOTE ON K-THEORY AND TRIANGULATED DERIVATORS
FERNANDO MURO AND GEORGE RAPTIS
Abstract. In this paper we show an example of two differential graded algebras that have the same derivator K-theory but non-isomorphic Waldhausen
K-theory. We also prove that Maltsiniotis’s comparison and localization conjectures for derivator K-theory cannot be simultaneously true.
1. Introduction
Derivators were introduced by Grothendieck [Gro90] as an intermediate step between a (suitable) category W with weak equivalences and its homotopy category
Ho(W ), obtained by formally inverting the weak equivalences. In addition to the
homotopy category Ho(W ), the derivator D(W ) contains all the homotopy cateop
gories of diagrams Ho(W I ) indexed by the opposite of a directed finite category I
together with the data of the various derived functors associated to the change of
indexing category. Formally, a derivator D is a 2-functor
D : Dirop
f −→ Cat
op
satisfying a set of axioms inspired by the example D(W )(I) = Ho(W I ). Here Cat
is the 2-category of small categories and Dirf ⊂ Cat is the 2-full sub-2-category of
directed finite categories.
Maltsiniotis added an extra axiom to define triangulated derivators [Mal07]. An
important example that was worked out by Keller [Kel07] is the derivator of the
category W = C b (E ) of bounded complexes in an exact category E , where the weak
equivalences are the quasi-isomorphisms in the abelian envelope of E . This derivator will be denoted D(E ) instead of D(C b (E )). Maltsiniotis defined the K-theory
K(D) of a triangulated derivator D, and stated three conjectures: comparison, localization and additivity. The proof of the additivity conjecture by Cisinski and
Neeman [CN08] showed that Maltsiniotis’s K-theory produces a K-theoretically
sound construction. The combination of the other two conjectures claims that
derivator K-theory, as a functor from a suitably defined category of small triangulated derivators to spaces, recovers Quillen’s K-theory on exact categories and
satisfies localization. Schlichting [Sch02] showed that there cannot exist a functor
from small triangulated categories to spaces that has these two properties.
1991 Mathematics Subject Classification. 19D99, 18E30, 16E45.
Key words and phrases. K-theory, exact category, Waldhausen category, Grothendieck
derivator.
The first author was partially supported by the Spanish Ministry of Education and Science
under the MEC-FEDER grant MTM2007-63277 and by the Government of Catalonia under the
grant SGR-119-2009.
1
2
FERNANDO MURO AND GEORGE RAPTIS
The comparison conjecture says that a certain natural map from Quillen’s to
Maltsiniotis’s K-theory space,
ρ : K(E ) −→ K(D(E )),
is a weak homotopy equivalence for every exact category E , i.e., the induced homomorphisms ρn = πn (ρ),
ρn : Kn (E ) −→ Kn (D(E )),
are isomorphisms for every n ≥ 0. Maltsiniotis proved this for n = 0 [Mal07]. The
conjecture is also true for n = 1 [Mur08]. Moreover, the following more general
version holds. If W is a Waldhausen category with cylinders and a saturated class
of weak equivalences, then D(W ) is a right pointed derivator [Cis10]. Garkusha
[Gar06] considered the derivator K-theory of this wider class of derivators. There
is a natural comparison map
(1.1)
µ : K(W ) −→ K(D(W )),
from Waldhausen’s to Garkusha’s K-theory, and so natural comparison homomorphisms µn = πn (µ), n ≥ 0,
µn : Kn (W ) −→ Kn (D(W )),
such that Maltsiniotis’s comparison map is obtained by precomposing with the
Gillet–Waldhausen equivalence [Gil81, TT90, Cis02],
∼
µ
ρ : K(E ) −→ K(C b (E )) −→ K(D(E )).
The homomorphisms µn are isomorphisms for n = 0, 1 [Mur08].
Toën and Vezzosi showed in [TV04] that µ cannot be an equivalence for all W
in all higher dimensions. More precisely, taking W = Rf (X) to be the category of
finite CW -complexes Y relative to a space X equipped with a retraction Y → X
[Wal85, 2.1], they noticed that the natural comparison map K(W ) → K(D(W ))
cannot be a weak homotopy equivalence in general, since the action of the topological monoid of self-equivalences of X on K(W ) does not factor through an aspherical
space. Despite the fact that the derivator encodes a lot of the homotopy theory
of W , it still forgets the higher homotopical information of the self-equivalences
of W , and in some sense that is exactly everything that it forgets [Ren09].
Thus the failure of the generalized comparison conjecture in this case is due to
the homotopical complexity of the algebraic K-theory of spaces. This could suggest
that it may however be true if we restrict to categories of algebraic origin such as,
for instance, the category W = Perf(A) of perfect right modules over a differential
graded algebra A. Note that D(Perf(A)) is a triangulated derivator while D(Rf (X))
is not.
Our first theorem shows that the generalized comparison conjecture is false even
in this algebraic context. Moreover, it shows that Kn (Perf(A)) need not be isomorphic to Kn (D(Perf(A))) even abstractly as abelian groups. This question was
so far open.
Let A = Fp [x±1 ] be the graded algebra of Laurent polynomials with a degree −1
indeterminate x, |x| = −1, over the field with p elements, p > 3 a prime number,
endowed with the trivial differential d = 0.
A NOTE ON K-THEORY AND TRIANGULATED DERIVATORS
3
Let B be the graded algebra over the integers Z generated by e and x±1 in degree
|e| = |x| = −1 with the following additional relations,
e2 = 0,
ex + xe = x2 ,
endowed with the degree 1 differential d defined by
d(e) = p,
d(x) = 0.
Notice that H ∗ (B) = A.
Theorem 1.2. The differential graded algebras A and B have equivalent derivator
K-theory,
K(D(Perf(A))) ' K(D(Perf(B))),
but different Waldhausen K-theory in dimension four K4 (Perf(A)) K4 (Perf(B)).
This is something close, although not actually a counterexample to Maltsiniotis’s original comparison conjeture. Nevertheless, we succeed in showing that the
comparison conjecture cannot be true if the localization conjecture holds. The localization conjecture says that any pair of composable pseudo-natural transformations
between triangulated derivators
D0 −→ D −→ D00
that yields a short exact sequence of triangulated categories D0 (I) → D(I) → D00 (I)
at any directed finite category I, induces a homotopy fibration in K-theory spaces,
K(D0 ) −→ K(D) −→ K(D00 ).
Theorem 1.3. Derivator K-theory does not satisfy both the comparison and localization conjectures.
2. The comparison homomorphisms
Let M be a stable pointed model category [Hov99]. An object X in the triangulated category T = Ho(M ) is compact if the functor T (X, −) preserves coproducts.
In most cases, the Waldhausen category W will essentially be the full subcategory
of compact cofibrant objects in some M . For any category I we equip the diagram
category M I with the model structure where weak equivalences and fibrations are
defined levelwise.
The poset n = {0 < 1 < · · · < n} is the free category generated by the following
Dynkin quiver of type An+1 ,
0 → 1 → · · · → n.
Qn =
A compact cofibrant object in M is a sequence of cofibrations between compact
cofibrant objects in M ,
X0 X1 · · · Xn .
Objects in the category Sn+1 W are such sequences together with choices of cofibers
Xij = Xj /Xi , 0 ≤ i ≤ j ≤ n, such that Xii = 0 is the distinguished zero object
of W . The morphisms are defined as commutative diagrams in the obvious way.
It is clear that Sn+1 W is equivalent to the full subcategory of M n that consists
of the compact cofibrant objects, the equivalence being the functor forgetting the
choice of cofibers. The categories Ho(W n ) and Ho(Sn+1 W ) are also equivalent.
For n = 0, S0 W is the category with one object 0 and one morphism (the identity).
These categories assemble to a simplicial Waldhausen category S• W where di+1
n
4
FERNANDO MURO AND GEORGE RAPTIS
and si+1 are induced by the usual functors di : n − 1 → n and si : n + 1 → n,
0 ≤ i ≤ n, the degeneracy s0 adds 0 at the beginning of the sequence,
0 X0 X1 · · · Xn ,
and the face d0 uses the choice of cofibers
X1 /X0 · · · Xn /X0 .
Waldhausen’s K-theory space of W is defined to be the loop space of the geometric realization of the simplicial subcategory of weak equivalences [Wal85],
K(W ) = Ω|wS• W |.
In most cases of interest, the Waldhausen category W produces a right pointed
derivator D(W ) [Cis10]. The K-theory space of D(W ) is naturally weakly equivalent
to
K(D(W )) ' Ω| iso(Ho(S• W ))|,
where iso(−) denotes the subcategory of isomorphisms, compare [Gar05, Mur08].
The natural comparison map (1.1) is induced by the canonical simplicial functor
S• W → Ho(S• W ), which takes weak equivalences to isomorphisms, and therefore
gives rise to a map
µ : Ω|wS• W | −→ Ω| iso(Ho(S• W ))|.
3. The proof of Theorem 1.2
For any commutative Frobenius ring R we denote Mod(R) the stable category of
R-modules, i.e., the quotient of the category Mod(R) of R-modules by the ideal of
morphisms which factor through a projective object. We denote MR the stable
pointed model category structure on Mod(R) where fibrations are the epimorphisms, cofibrations are the monomorphisms, and weak equivalences are those
homomorphisms which project to an isomorphism in Mod(R) [Hov99, Theorem
2.2.12]. The homotopy category is Ho(MR ) = Mod(R) and the compact objects
are essentially the finitely generated R-modules. We denote WR the Waldhausen
subcategory of finitely generated R-modules, whose underlying category is denoted
by mod(R). Its homotopy category Ho(WR ) = mod(R) is the full subcategory of
Mod(R) spanned by the finitely generated R-modules.
In this section we will compute Ho(WRn ) for R a commutative noetherian local
ring with principal maximal ideal (α) 6= 0 such that α2 = 0. We denote k = R/α
the residue field. The upshot of the computation will be that the category Ho(WRn )
is determined up to equivalence by the field k.
The first important observation, for n = 0, is that the full inclusion of the
category of finite dimensional k-vector spaces mod(k) ⊂ mod(R) induced by the
unique ring homomorphism q : R k gives rise to an equivalence of categories
∼
mod(k) −→ mod(R).
For any n ≥ 0, this functor induces a section of the canonical functor Ho(WRn ) →
Ho(WR )n , since the following composite is an equivalence,
(3.1)
mod(k)n ⊂ mod(R)n = WRn −→ Ho(WRn ) −→ Ho(WR )n = mod(R)n .
Proposition 3.2. The additive functor Ho(WRn ) → Ho(WR )n is full, essentially
surjective, and reflects isomorphisms. In particular it induces a bijection between
the corresponding monoids of isomorphism classes of objects.
A NOTE ON K-THEORY AND TRIANGULATED DERIVATORS
5
Proof. By [Cis10, Proposition 2.15] the functor is full and essentially surjective. It
reflects isomorphisms since weak equivalences in WR satisfy the strong saturation
property.
The category mod(k)n is well known since it is the category of finite-dimensional
representations of the quiver Qn over the field k. The monoid of isomorphism classes
of objects is freely generated by the following set of indecomposable representations,
En = {Mi,j ; 0 ≤ i ≤ j ≤ n},
1
1
Mi,j = |0 → ·{z
· · → 0} → k
· · → k} → 0| → ·{z
· · → 0} .
| → ·{z
i
j−i+1
n−j
Given a category C and a set of objects E in C we denote CE ⊂ C the full
subcategory spanned by the objects in E. We now describe mod(k)n
En . Consider
the poset
Pn = {(i, j) ; 0 ≤ i ≤ j ≤ n} ⊂ Z × Z
and the corresponding k-linear category k[Pn ]. We refer to [Mit72] for the basics
on the ring and module theory of linear categories. Then there is an isomorphism
of k-linear categories
∼
=
k[Pn ]op /((i − 1, i − 1) ≤ (i, i))ni=1 −→ mod(k)n
En
sending (i, j) to Mi,j and (i1 , j1 ) ≤ (i2 , j2 ) to the morphism Mi2 ,j2 → Mi1 ,j1 given
by
1
1
1
1
1
0 → ··· → 0 → 0 → ··· → 0 → k → ··· → k → k → ··· → k → 0 → ··· → 0
↓
↓
↓
↓
↓1
↓1 ↓
↓
↓
↓
1
1
1
1
1
0 → ··· → 0 → k → ··· → k → k → ··· → k → 0 → ··· → 0 → 0 → ··· → 0
Equivalently, mod(k)n
En is isomorphic to the k-linear category of the following
quiver ΓQn
(0, n)
?
??
??


(0, n − 1)
(1, n)
?
??
??


(1, n − 1)
(i − 1, j)
??
?
??


(i − 1, j − 1)
(i, j)
??
?

??


(i, j − 1)
(n − 1, n)
?
??
??


(n − 1, n − 1)
(n, n)
(i − 1, i)
?
??
??


(i − 1, i − 1)
(i, i)
(0, 1)
?
??
??


(0, 0)
(1, 1)
6
FERNANDO MURO AND GEORGE RAPTIS
e.g. for n = 3
ΓQ3
(0, 3)
?
??
??



(1, 3)
(0, 2)
?
?
??
??
??
??




(2, 3)
(1, 2)
(0, 1)
?
??
??
?
??
?
??
??
??









(3, 3)
(2, 2)
(1, 1)
(0, 0)
=
with relations:
• The square
(i − 1, j)
?
??
??


(i, j)
(i − 1, j − 1)
??
?
??


(i, j − 1)
• The composite
is commutative for 0 < i < j ≤ n,
(i − 1, i)
?
??
??



(i − 1, i − 1)
(i, i)
is trivial for 0 < i ≤ n.
This category will be denoted by kτ ΓQn since it is (isomorphic to) the mesh category
of the quiver ΓQn with translation τ (i, j) = (i + 1, j + 1), 0 ≤ i ≤ j < n, which
is the Auslander-Reiten quiver of Qn , see [BG82]. The arrows % and & in ΓQn
correspond to the following morphisms,
1
(3.3)
% =
& =
0 → ··· → 0 → k → ··· → k → k → 0 → ··· → 0
↓
↓
↓1
↓1 ↓
↓
↓
1
1
0 → ··· → 0 → k → ··· → k → 0 → 0 → ··· → 0
1
(3.4)
1
0 → ··· → 0 → 0 → k → ··· → k → 0 → ··· → 0
↓
↓
↓
↓1
↓1 ↓
↓
1
1
1
0 → ··· → 0 → k → k → ··· → k → 0 → ··· → 0
1
1
Recall that given a k-linear category C and a C -bimodule D the semidirect
product D o C is the category with the same objects as C , morphism k-modules
(D o C )(X, Y ) = D(X, Y ) ⊕ C (X, Y ),
and composition defined by
(b, g)(a, f ) = (b · f + g · a, gf ).
Let Dn be the kτ ΓQn -bimodule with generators
ϕi ∈ Dn ((0, i − 1), (i, n)),
0 < i ≤ n,
and relations
((i + 1, n) → (i, n)) · ϕi+1 = ϕi · ((0, i) → (0, i − 1)),
((1, n) → (0, n)) · ϕ1 = 0.
0 < i < n;
A NOTE ON K-THEORY AND TRIANGULATED DERIVATORS
7
This kτ ΓQn -bimodule is
Dn ((i1 , j1 ), (i2 , j2 )) ∼
= k,
i1 + 1 ≤ i2 ≤ j1 + 1 ≤ j2 ,
generated by
((i2 , n) → (i2 , j2 )) · ϕi2 · ((i1 , j1 ) → (0, i2 − 1)),
and zero otherwise. It is a well known fact from representation theory that, under
∼
the isomorphism kτ ΓQn ∼
= mod(k)n
En , we have a bimodule isomorphism Dn =
1
Extmod(k)n sending ϕi to the element represented by the following extension
0 → ··· →
↓
1
1
k → ··· →
↓1
1
1
k → ··· →
1
1
0 → k → ··· → k
↓
↓1
↓1
1
1
1
k → k → ··· → k
↓1 ↓
↓
k → 0 → ··· → 0
compare [ARS95], hence
Dn o kτ ΓQn ∼
= Ext1mod(k)n omod(k)n
En .
∼
For 0 ≤ i ≤ j ≤ n we consider the cofibrant replacement ri,j : M̃i,j Mi,j of
Mi,j in WRn given by
1
1
α
1
1
0 → ··· → 0 → k → ··· → k → R → ··· → R
↓
↓
↓1
↓1 ↓
↓
1
1
0 → ··· → 0 → k → ··· → k → 0 → ··· → 0
Notice that ri,n is the identity, 0 ≤ i ≤ n.
Proposition 3.5. There is an isomorphism of categories
∼
=
Dn o kτ ΓQn −→ Ho(WRn )En
which is defined on the second factor by the isomorphism kτ ΓQn ∼
= mod(k)n
En and
the splitting (3.1), and on the first factor it sends ϕi , 0 < i ≤ n, to the inverse of
r0,i−1 in the homotopy category composed with the morphism ϕ̃i : M̃0,i−1 → Mi,n
below,
1
1
α
1
1
k → ··· → k → R → ··· → R
↓
↓
↓q
↓q
1
1
0 → ··· → 0 → k → ··· → k
Proof. The arrows Mi,j → Mi−1,j and Mi,j → Mi,j−1 in (3.3) and (3.4) lift to
the following two morphisms M̃i,j → M̃i−1,j and M̃i,j → M̃i,j−1 between cofibrant
replacements, respectively,
1
1
α
1
1
α
1
1
(3.6)
0 → ··· → 0 → 0 → k → ··· → k → R → ··· → R
↓
↓
↓
↓1
↓1 ↓1
↓1
1
1
1
α
1
1
0 → ··· → 0 → k → k → ··· → k → R → ··· → R
(3.7)
0 → ··· → 0 → k → ··· → k → k → R → ··· → R
↓
↓
↓1
↓1 ↓α ↓1
↓1
1
1
α
1
1
1
0 → ··· → 0 → k → ··· → k → R → R → ··· → R
1
1
1
Given (i1 , j1 ) ≥ (i2 , j2 ) we denote
νi1 ,j1 ;i2 ,j2 : Mi1 ,j1 −→ Mi2 ,j2 ,
ν̃i1 ,j1 ;i2 ,j2 : M̃i1 ,j1 −→ M̃i2 ,j2 ,
8
FERNANDO MURO AND GEORGE RAPTIS
the unique composite of (3.3)’s and (3.4)’s, and of (3.6)’s and (3.7)’s, respectively.
We have ri2 ,j2 ν̃i1 ,j1 ;i2 ,j2 = νi1 ,j1 ;i2 ,j2 ri1 ,j1 .
One can easily check by inspection that the epimorphism ri1 ,j1 induces the
monomorphism
mod(R)n (Mi1 ,j1 , Mi2 ,j2 ) ,→ mod(R)n (M̃i1 ,j1 , Mi2 ,j2 )
given by:
• k ,→ k, an isomorphism, if i2 ≤ i1 ≤ j2 ≤ j1 ;
• 0 ,→ k if i2 ≤ j1 + 1 ≤ j2 , with target generated by
(3.8)
νi2 ,n;i2 ,j2 ϕ̃i2 ν̃i1 ,j1 ;0,i2 −1 ,
if i1 + 1 ≤ i2 ; and if i1 ≥ i2 by
(3.9)
1
1
α
1
1
1
1
1
0 → ··· → 0 → 0 → ··· → 0 → k → ··· → k → R → ··· → R → R → ··· → R
↓
↓
↓
↓
↓0
↓0 ↓q
↓q ↓
↓
1
1
1
1
1
1
1
1
0 → ··· → 0 → k → ··· → k → k → ··· → k → k → ··· → k → 0 → ··· → 0
• 0 ,→ 0 otherwise.
The morphism (3.9) is trivial in Ho(WRn ) since it factors through a contractible
object as follows,
0 → ··· →
↓
0 → ··· →
↓
0 → ··· →
0 → 0 → ··· →
↓
↓
0 → 0 → ··· →
↓
↓
1
1
0 → k → ··· →
1
1
α
1
1
1
1
1
0 → k → ··· → k → R → ··· → R → R → ··· → R
↓
↓α
↓α ↓1
↓1 ↓1
↓1
1
1
1
1
1
1
1
1
0 → R → ··· → R → R → ··· → R → R → ··· → R
↓
↓q
↓q ↓q
↓q ↓
↓
1
1
1
1
1
1
k → k → ··· → k → k → ··· → k → 0 → ··· → 0
On the contrary (3.8) cannot be trivial in Ho(WRn ) since it contains a commutative
square of the form
α
k→R
↓
↓q
0→ k
which induces the non-trivial morphism 1 : k → k in Ho(WR ) ' mod(k) on cofibers
(i.e. cokernels) of horizontal arrows (which are cofibrations).
These computations prove that the kernel of Ho(WRn )En → Ho(WR )n
En is the
ideal In with
In (Mi1 ,j1 , Mi2 ,j2 ) ∼
= k,
i1 + 1 ≤ i2 ≤ j1 + 1 ≤ j2 ,
generated by the inverse of ri1 ,j1 composed with (3.8); and zero otherwise.
In order to prove that the functor in the statement is well defined we only have
to check that In is a square-zero ideal. The only non-trivial cases to check are the
composites of generators
Mi1 ,j1 −→ Mi2 ,j2 −→ Mi3 ,j3 ,
i1 + 3 ≤ i2 + 2 ≤ i3 + 1 ≤ j1 + 2 ≤ j2 + 1 ≤ j3 .
Such a composite, precomposed with the isomorphism ri1 ,j1 in the homotopy category, is
νi3 ,n;i3 ,j3 ϕ̃i3 ν̃i2 ,j2 ;0,i3 −1 ν̃i2 ,n;i2 ,j2 ϕ̃i2 ν̃i1 ,j1 ;0,i2 −1
= νi3 ,n;i3 ,j3 ϕ̃i3 ν̃i2 ,n;0,i3 −1 ϕ̃i2 ν̃i1 ,j1 ;0,i2 −1 .
A NOTE ON K-THEORY AND TRIANGULATED DERIVATORS
9
The morphism ν̃i2 ,n;0,i3 −1 is
1
1
1
1
1
0 → ··· → 0 → k → ··· → k → k → ··· → k
↓
↓
↓1
↓1 ↓α
↓α
1
1
1
1
1
α
1
1
k → ··· → k → k → ··· → k → R → ··· → R
therefore ϕ̃i3 ν̃i2 ,n;0,i3 −1 = 0, and hence also the composite Mi1 ,j1 → Mi3 ,j3 .
Now that we have proved that the functor is well defined, the previous computation of In automatically shows that the functor is actually an isomorphism,
see the descriptions of the bimodule Dn . Moreover, Ho(WRn )En → Ho(WR )n
En and
its splitting correspond to the projection and the inclusion of the second factor of
Dn o kτ ΓQn , respectively.
For any k-linear category C we denote C+ the k-linear category obtained by
formally adding a zero object 0. This is the left adjoint of the forgetful functor
from k-linear categories with zero object to k-linear categories.
We define the simplicial k-linear category with zero object D• (k) as D0 (k) = 0,
Dn+1 (k) = (Dn o kτ ΓQn )+ , n ≥ 0, faces and degeneracies defined on objects by

(i − 1, j − 1), 0 ≤ t < i ≤ j ≤ n;



(i, j − 1),
0 ≤ i ≤ t ≤ j ≤ n, i < j;
dt+1 (i, j) =
0,
0 ≤ i = t = j ≤ n;



(i, j),
0 ≤ i ≤ j < t ≤ n;

 (j + 1, n − 1), 0 = i ≤ j < n;
0,
0 = i ≤ j = n;
d0 (i, j) =

(i − 1, j − 1), 0 < i ≤ j ≤ n;

 (i + 1, j + 1), −1 ≤ t < i ≤ j ≤ n;
(i, j + 1),
0 ≤ i ≤ t ≤ j ≤ n;
st+1 (i, j) =

(i, j),
0 ≤ i ≤ j < t ≤ n;
on morphisms in kτ ΓQn as
0,
if dt (i1 , j1 ) = 0 or dt (i2 , j2 ) = 0;
dt ((i1 , j1 ) ≥ (i2 , j2 )) =
dt (i1 , j1 ) ≥ dt (i2 , j2 ), otherwise;
st ((i1 , j1 ) ≥ (i2 , j2 )) = st (i1 , j1 ) ≥ st (i2 , j2 );
and on generators of Dn as

t = 0, i = 1;
 0,
ϕi−1 , 0 ≤ t < i ≤ n, i > 1;
dt+1 (ϕi ) =

ϕi ,
0 < i ≤ t ≤ n;
d0 (ϕi ) = 1(i−1,n−1) , 0 < i ≤ n;
ϕi+1 , 0 ≤ t < i ≤ n;
st+1 (ϕi ) =
ϕi ,
0 < i ≤ t ≤ n;
s0 (ϕi ) = ϕi+1 · ((1, i) → (0, i)),
0 < i ≤ n.
We leave the reader the tedious but straightforward task to check that D• (k) is well
defined. Notice that this simplicial category only depends on the field k. Actually,
it is functorial in k. Moreover, the inclusion of the second factors
En (k) := (kτ ΓQn )+ ⊂ Dn (k)
10
FERNANDO MURO AND GEORGE RAPTIS
define a simplicial k-linear subcategory with zero object E• (k) ⊂ D• (k), however
the face d0 does not take all elements in Dn to elements in Dn−1 .
Consider the following set of objects of Sn+1 WR ,
Ẽn = {M̃i,j ; 0 ≤ i ≤ j ≤ n},
n ≥ 0;
Ẽ−1 = ∅;
with the choices of cofibers indicated in the following short exact sequences,
(3.10)
1
0 ,→ 0 0,
1
0 ,→ k k,
1
α
0 ,→ R R,
k ,→ k 0,
q
1
k ,→ R k,
R ,→ R 0.
Note that the full inclusions Ho(Sn WR )Ẽn−1 ∪{0} ⊂ Ho(Sn WR ) define a simplicial
subcategory with zero object Ho(S• WR )Ẽ•−1 ∪{0} ⊂ Ho(S• WR ).
Lemma 3.11. There is a unique isomorphism
∼ Ho(S• WR )
ξ(R, α) : D• (k) =
Ẽ•−1
such that the cofibrant replacements ri,j : M̃i,j Mi,j define natural isomorphisms
indicated with double arrow in the following diagram, 0 ≤ i ≤ j ≤ n,
Dn+1 (k)
(3.5) ∼
=
ξ(R,α)n+1
Ho(WRn )En ∪{0}
∼
=
/ Ho(Sn+1 WR )Ẽ
uu
v~ uuuu∼
=
n ∪{0}
full inclusion
full inclusion
/ Ho(W n )
R
This lemma is just an observation. Actually, this observation motivated our
definition of D• (k).
The additivization of a k-linear category C is the category C add whose objects are
sequences (C1 , . . . , Cn ) of finite length n ≥ 0 of objects in C , and whose morphisms
are matrices,
M
C add ((Ci )ni=1 , (Cj0 )m
C (Ci , Cj0 ).
j=1 ) =
1≤i≤n
1≤j≤m
This is an additive k-linear category with zero object () the empty sequence (of
legth n = 0). The additivization is the left adjoint (in the weak 2-categorical sense)
of the forgetful 2-functor from small additive k-linear categories to small k-linear
linear categories (weak because the counit is given by the choice of direct sums
C1 ⊕ · · · ⊕ Cn and zero object 0 in C ). Moreover, if E is a set of generators of the
∼
monoid of isomorphism classes of objects of C then an adjoint CEadd → C of the
full inclusion CE ⊂ C is an equivalence.
Corollary 3.12. The isomorphism ξ(R, α) in Lemma 3.11 induces a simplicial
k-linear functor
∼
˜ α) : D• (k)add −→
ξ(R,
Ho(S• WR )
which is a levelwise equivalence. In particular, restricting to isomorphisms we obtain a simplicial functor which is a levelwise equivalence,
∼
˜ α)) : iso(D• (k)add ) −→ iso(Ho(S• WR )).
iso(ξ(R,
Now we are ready to prove Theorem 1.2.
A NOTE ON K-THEORY AND TRIANGULATED DERIVATORS
11
Proof of Theorem 1.2. Let p > 3 be a prime. The rings Fp [ε]/ε2 and Z/p2 are
commutative noetherian local with non-trivial maximal ideal generated by α = ε
and p, respectively, and such that α2 = 0, therefore they satisfy the assumptions of
this section.
Dugger and Shipley proved that MFp [ε]/ε2 and MZ/p2 are Quillen equivalent to
the categories Mod(A) and Mod(B) of right modules over the diferential graded algebras defined in the intorduction, respectively, where weak equivalences are quasiisomorphisms and fibrations are levelwise surjections [DS09, Corollary 4.4] (we
here use cohomological degree, unlike Dugger and Shipley). This, together with
the approximation theorem in K-theory (e.g. see [Cis09, Théorème 2.15], [DS04,
Corollary 3.10]) shows that we can replace everywhere Perf(A) and Perf(B) with
WFp [ε]/ε2 and WZ/p2 , respectively.
Now the equivalence in the statement of the theorem is obtained by taking the
loop space of the geometric realization of the second map in the previous corollary.
The last claim is a computation made in [Sch02, 1.6], see also [DS09, Remark
4.9].
Remark 3.13. Note that the approximation theorem in K-theory implies that the
stable model categories MFp [ε]/ε2 and MZ/p2 cannot be connected by a zigzag of
Quillen equivalences. By a result of Renaudin [Ren09, Théorème 3.3.2], it follows
that the associated triangulated derivators are not equivalent. However, by Corollary 3.12, their difference cannot be detected on the part that is involved in the
definition of derivator K-theory.
4. The proof of Theorem 1.3
Given a commutative noetherian ring R, let mod(R) be the abelian (in particular
exact) category of finitely generated R-modules and j : proj(R) → mod(R) the
inclusion of the full exact subcategory of projectives. There is an induced morphism
between the associated triangulated derivators,
D(j) : D(proj(R)) −→ D(mod(R)),
which is given for every I in Dirf by the inclusion of a thick subcategory,
op
op
D(j)(I) : Db (proj(R)I ) −→ Db (mod(R)I ).
The pointwise Verdier quotients define a prederivator,
D(R) : Dirop
f −→ Cat,
op
op
I 7→ Db (mod(R)I )/Db (proj(R)I ).
Let e denote the terminal category and, given A in Dirf and an object a in A, let
iA,a : e → A be the inclusion of a.
Proposition 4.1. For every directed finite category A, the collection of functors
i∗A,a : D(R)(A) → D(R)(e), a in A, is conservative, i.e., a morphism f in D(R)(A)
is an isomorphism if and only if i∗A,a (f ) is an isomorphism for every a in A.
In the proof of this proposition we will use the following lemma.
Lemma 4.2. Let P ∗ be a bounded above complex of projective R-modules which is
quasi-isomorphic to a bounded complex of projective R-modules Q∗ with Qm = 0
for m ≤ N . Then Im[dm : P m−1 → P m ] = Ker[dm+1 : P m → P m+1 ] is a projective
R-module for all m < N .
12
FERNANDO MURO AND GEORGE RAPTIS
Proof. Both P ∗ and Q∗ are fibrant and cofibrant in the model category C(Mod(R))
of complexes of R-modules [Hov99, Lemma 2.3.6 and Theorem 2.3.11], therefore
∼
there exists a quasi-isomorphism f : Q∗ → P ∗ . The mapping cone Cf∗ of f is
a bounded above acyclic complex of projective R-modules, therefore Cf∗ is contractible [Hov99, Lemma 2.3.8], in particular
Im[dm : Cfm−1 → Cfm ] = Ker[dm+1 : Cfm → Cfm+1 ]
is a projective R-module for all m ∈ Z. The complex Cf∗ coincides with P ∗ in
degrees m < N , hence we are done.
We can now prove the proposition.
Proof of Proposition 4.1. Let f : X → Y be a morphism in D(R)(A) such that
i∗A,a (f ) is an isomorphism for all objects a in A. Let
f˜: X̃ −→ Ỹ
be a morphism in D(mod(R))(A) whose image under the localization functor
S : D(mod(R))(A) −→ D(R)(A)
is isomorphic to f . Since S is a triangulated functor, the distinguished triangle
f˜
X̃ −→ Ỹ −→ Z̃ −→ X̃[1]
op
in D(mod(R)A ) maps to a distinguished triangle isomorphic to
f
X −→ Y −→ Z −→ X[1]
and so, by the assumption, there is an object Pa in Db (proj(R)) and an isomorphism
ga : Pa ∼
= i∗A,a (Z̃) for every object a in A. Let N be an integer such that Pam = 0
for every m ≤ N and a in A.
We consider the closed model category, in the sense of Quillen [Qui67], of bounded
above complexes C − (mod(R)) of finitely generated R-modules, where the weak
equivalences are the quasi-isomorphisms and the fibrations are the levelwise surop
jective morphisms. In the derivable category C − (mod(R))A [Cis10], there is a
∼
cofibrant resolution given by a pointwise quasi-isomorphism Q̃ → Z̃ where Q̃ is a
diagram of bounded above complexes of finitely generated projective R-modules.
By the previous lemma, replacing the complex
dN −2
dN −1
dN
dn
dN
dn
−3
−2
−1
Q̃a = · · · → Q̃N
−→ Q̃N
−→ Q̃N
→ · · · → Q̃an−1 → Q̃na → · · · → 0 → · · ·
a
a
a
with its functorial truncation
−1
τ≥N −1 Q̃a = · · · → 0 → Ker(dN ) → Q̃N
→ · · · → Q̃n−1
→ Q̃na → · · · → 0 → · · ·
a
a
gives a diagram of bounded complexes of finitely generated projective R-modules
∼
such that the inclusion τ≥N −1 Q̃ → Q̃ is a pointwise quasi-isomorphism. Hence
op
op
Z̃ is isomorphic in Db (mod(R)A ) to an object in Db (proj(R)A ), so S(f˜) is an
isomorphism, as required to show.
Let W˜R be the Waldhausen category with cylinders with underlying category
C (mod(R)) where cofibrations are the monomorphisms and weak equivalences are
the morphisms whose mapping cone is quasi-isomorphic to a bounded complex of
finitely generated projective R-modules. That is precisely the strongly saturated
b
A NOTE ON K-THEORY AND TRIANGULATED DERIVATORS
13
class of morphisms that map into isomorphisms under the functor C b (mod(R)) →
Db (mod(R))/Db (proj(R)).
Proposition 4.3. There is an isomorphism of prederivators D(W˜R ) ∼
= D(R). In
particular, D(R) is a triangulated derivator.
Proof. Let I be a directed finite category. The universal property of localizations
implies that the dotted arrows can be uniquely filled in the following commutative
diagram
op
C b (mod(R)I )
/ Db (mod(R)I op )/Db (proj(R)I op ).
/ Db (mod(R)I op )
vr
op
Ho(W˜RI )
Similarly in the following diagram, using Proposition 4.1,
/ Db (mod(R)I op )/Db (proj(R)I op )
4
op
C b (mod(R)I )
op
Ho(W˜RI )
By the universal property, the functors
op
op
op
Ho(W˜ I ) Db (mod(R)I )/Db (proj(R)I )
R
are mutually inverse isomorphisms, natural in I, hence the result follows.
Proposition 4.4. Let R be a commutative Frobenius ring. The full inclusion
G : WR = mod(R) → C b (mod(R)) = W˜R of complexes concentrated in degree 0
induces an equivalence of triangulated derivators
∼
D(G) : D(WR ) −→ D(W˜R ).
Proof. By [Ric89, Theorem 2.1], G induces an equivalence
∼
Ho(G) : Ho(WR ) = mod(R) −→ Db (mod(R))/Db (proj(R)) ∼
= Ho(W˜R ).
The functor G is a right exact functor between strongly saturated right derivable
categories, therefore by [Cis10, Théorèm 3.19] it induces an equivalence D(G) of
triangulated derivators.
We are now ready to prove our second theorem.
Proof of Theorem 1.3. As in the previous section, let R be a commutative noetherian local ring with principal maximal ideal (α) 6= 0 such that α2 = 0, k = R/α
the residue field, and q : R k the natural projection. We consider the full simplicial subcategory B• (k) ⊂ E• (k) spanned by the zero object and (i, n), 0 ≤ i ≤ n.
The following set En0 is a basis of the monoid of isomorphism classes of objects in
Sn+1 (mod(k)),
En0 = {Mi,n ; 0 ≤ i ≤ n},
1
n ≥ 0,
1
· · → k},
Mi,n = |0 → ·{z
· · → 0} → k
| → ·{z
i
n−i+1
0
E−1
= ∅,
14
FERNANDO MURO AND GEORGE RAPTIS
with choices of cofibers as in (3.10). There is defined an isomorphism of simplicial
0
categories preserving the zero object B• (k) ∼
= S• (mod(k))E•−1
∪{0} : (i, n) 7→ Mi,n
sending (i, n) → (i − 1, n) to
1
1
0 → ··· → 0 → 0 → k → ··· → k
↓
↓
↓
↓1
↓1
1
1
1
0 → ··· → 0 → k → k → ··· → k
∼
This isomorphism defines a levelwise equivalence of simplicial categories B• (k)add →
S• (mod(k)) fitting into a commutative diagram
B• (k) add
_
induced by q
E• (k) add
_
D• (k)add
/ S• (mod(k))
∼
S• (mod(R))
complexes
concentrated
in degree 0
/ Ho(S• (C b (mod(R))))
canonical
∼
ξ̃(R,α)
/ Ho(S• (WR ))
∼
Proposition 4.4
/ Ho(S• (W˜R ))
Considering the subcategories of isomorphisms and taking the loop space of the
geometric realizations in this diagram, we obtain the following commutative diagram,
Ω| iso(B• (k)add )|
_
∼
/ K(mod(k))
∼ dévissage
K(mod(R))
Ω| iso(D• (k)add )|
∼
/ K(D(WR ))
/ K(D(mod(R)))
ρ
∼
/ K(D(R))
Props. 4.3 and 4.4
Assume that derivator K-theory satisfies both the localization and comparison conjectures. Then ρ is an equivalence and the short exact sequence of triangulated
derivators
D(proj(R)) −→ D(mod(R)) −→ D(R)
induces a homotopy fibration of K-theory spaces
K(D(proj(R))) −→ K(D(mod(R))) −→ K(D(R)),
ρ
therefore K(R) = K(proj(R)) ' K(D(proj(R))) is the homotopy fiber of the map
Ω| iso(B• (k)add )| → Ω| iso(D• (k)add )|,
which is independent of R (it only depends on k), but this contradicts known
calculations showing that, for p > 3, Fp [ε]/ε2 and Z/p2 have inequivalent Quillen
K-theories, more precisely K3 (Fp [ε]/ε2 ) K3 (Z/p2 ), compare [Sch02, 1.6] and
[DS09, Remark 4.9].
A NOTE ON K-THEORY AND TRIANGULATED DERIVATORS
15
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16
FERNANDO MURO AND GEORGE RAPTIS
Universidad de Sevilla, Facultad de Matemáticas, Departamento de Álgebra, Avda.
Reina Mercedes s/n, 41012 Sevilla, Spain
E-mail address: [email protected]
Universität Osnabrück, Institut für Mathematik, Albrechtstrasse 28a, 49069 Osnabrück, Germany
E-mail address: [email protected]